Measuring Chemical Likeness of Stars with Relevant Scaled Component Analysis

The characteristics within a stellar spectrum are caused by the interplay of many factors of variation. These include chemical and physical parameters of the star and the instrumental systematics associated with the telescope, as well as interstellar dust along the line of sight. Measuring chemical similarities requires disentangling the imprint left on the spectra by chemical factors of variation from that left by the other nonchemical factors of variation. Our goal is to identify chemically similar stars from their spectra, for stars that span a range of physical stellar parameters (i.e., effective temperatures and surface gravities). We approach this task from a data-driven perspective, and build an algorithm for identifying stars that are as chemically similar as birth siblings, using open-cluster spectra.

For our method, we assume that open clusters are close to chemically homogeneous because of their common birth origin (Ness et al. 2018) but are not special in any other way (at least in terms of their spectra). That is to say, we assume that the only information within spectra useful for recognizing open clusters are the chemical features of the spectra, and so that a model which identifies open clusters from the spectra will need to do so by extracting the chemical information within spectra.

We frame the task of building such a model recognizing open clusters as a metric-learning task. That is to say, we build a data-driven model converting stellar spectra into a representation in which Euclidean distances convey the uncalibrated probability of stars originating from a shared open cluster. To accomplish this, the training objective of our data-driven algorithm can be understood as transforming stellar spectra into a representation in which the distance between intracluster stars is minimized and the distance between intercluster stars is maximized.

Distances in the representation resulting from such an optimizing procedure will organically quantify the chemical similarity of stars. Nonchemical factors of variation, such as stellar temperatures and instrumental systematics, will not contribute to the representation as their presence would make distances between stellar siblings larger. Instead, such a representation will only contain those factors of variation of spectra that are discriminative of open clusters, i.e., the chemical factors of variation. Crucially, chemical factors of variation will contribute to distances in the representation in proportion to how precisely they can be estimated from stellar spectra. Stronger chemical features will be more strongly amplified than weaker chemical features.

The utility of this data-driven approach is that it is independent of imperfect model atmosphere approximations and other issues affecting synthetic spectra. This provides a high-fidelity technique to turn to specific applications within Galactic archeology, such as the chemical tagging of stars that are most chemically similar (Freeman & Bland-Hawthorn 2002).

The assumption underpinning our work is that this chemical information will be the only information within stellar spectra useful for distinguishing open clusters and so will be the only information captured by our model. If this assumption is true, then the representation induced by the model will measure a form of chemical similarity between stellar spectra.

However, since open clusters, in addition to sharing a common age and near-identical birth abundances, are also gravitationally bound, they can be identified from their spatial proximity if such information is available in the spectra. As such spatial information does not robustly transfer toward identifying dissolved clusters, it must not be captured by our model. In this work, we apply our algorithm to pseudo-continuum normalized spectra with diffuse interstellar bands masked, which we assume not to contain any information about spatial location so as for our representation after training to only contain chemical information. Assuming pseudo-continuum normalized spectra do not encode any spatial information is plausible, since after continuum normalization, the spectrum should not contain significant information about stellar distance. With the impact of reddening removed and diffuse interstellar bands masked, a spectrum should also not contain any information about the stellar extinction and interstellar medium along the line of sight of the star. We examine the validity of these assumptions in latter sections. Ultimately, it is worth emphasizing that our method only exploits features proportionally to their discriminativeness at recognizing open clusters. Therefore, we can expect our model to not heavily rely on nonrobust features (provided that these are significantly less informative than robust features). This also relies on our open-cluster training data being representative of the parameter space that should be marginalized out, i.e., our model does not learn to associate intercluster stars via $\mathrm{log}g$ and T_eff, which could happen if the evolutionary state of the observed cluster stars was similar within clusters and different between clusters.

Our metric-learning algorithm, RSCA, first uses PCA to transform the data into a (lower-dimensional) basis that represents the primary variability of the ensemble of spectra we work with.

The principal components of a data set X, of shape N_D × N_F containing N_D data points and N_F features, are an ordered orthogonal basis of the feature space with special properties. In the principal-component basis, basis vectors are ordered by the amount of variance they capture. They have the property that for any k, the hyperplane spanned by the first k-axes of the basis is the k-dimensional hyperplane, which maximally captures the data variance. In PCA, the number of principal components used, k, is a hyperparameter controlling the trade-off between the amount of information preserved in the data set X after compression and the degree of compression.

The principal-component basis corresponds to the unit-norm eigenvectors of the covariance matrix of X ordered by eigenvalue magnitude. This can be obtained through diagonalization of the covariance matrix. The principal-component basis can also be formulated as the maximum-likelihood solution of a probabilistic latent model, which is known as probabilistic principal component analysis (PPCA; see Bishop 2006). This probabilistic formulation is useful in that it enables one to obtain the principal components for a data set containing missing values by marginalizing over these.

As we will make use of further in this paper, the principal components also allow for generating a sphering transformation. This is a linear transformation of the data set to a new representation, in which the covariance matrix of the data set X is the identity matrix. Sphering using PCA is carried out by performing a change-of-basis to a modified principal-component basis in which the principal components are divided by the square root of their associated eigenvalues.

Let us define X_clust as the matrix representation of a data set containing the spectra of known open-cluster stars. Analogously, let us define X_pop as the matrix representation of a larger data set of stellar spectra in the field (with unknown cluster membership). These matrices are, respectively, of shapes ${N}_{{{\rm{d}}}_{\mathrm{clust}}}\times {N}_{{\rm{b}}}$ and ${N}_{{{\rm{d}}}_{\mathrm{pop}}}\times {N}_{{\rm{b}}}$, where ${N}_{{{\rm{d}}}_{\mathrm{clust}}}$ is the number of open-cluster stars, ${N}_{{{\rm{d}}}_{\mathrm{pop}}}$ the number of of stars in the large data set and N_b the number of spectral bins. For our purposes, we assume access to only a limited number of open-cluster stars such that ${N}_{{{\rm{d}}}_{\mathrm{clust}}}\ll {N}_{{{\rm{d}}}_{\mathrm{pop}}}$. We also assume that the spectra in these matrices are pseudo-continuum normalized spectra, with diffuse interstellar bands masked in a process following that described in Appendix A. Pseudo-continuum spectra are normalized rest-frame spectra in which the effects of interstellar reddening and atmospheric absorption are removed, in a process described in Majewski et al. (2017).

RSCA takes as inputs X_clust and X_pop. Through a series of linear transformation to these matrices, RSCA maps these matrices into new matrices of shape ${N}_{{{\rm{d}}}_{\mathrm{clust}}}\times {N}_{{\rm{K}}}$ and ${N}_{{{\rm{d}}}_{\mathrm{pop}}}\times {N}_{{\rm{K}}}$ whose entries are the stellar spectra transformed to a metric-learning representation of dimensionality K. Euclidean distances in this new metric-learning representation can then be used to measure chemical similarity between spectra. As all the steps of RSCA are linear transformations, the mapping converting from spectra to the metric-learning representation can be parameterized by an N_K × N_bins matrix and used to convert unseen spectra (or for visualization purposes).

We provide in Figure 1 a graphical depiction of the linear transformations involved in the RSCA algorithm. RSCA works by first projecting the spectra onto a set of basis vectors with PCA (Step 1). For visualization purposes this basis is made two-dimensional, although it would normally be higher dimensional. After this PCA compression (Step 1), stellar siblings are represented as same-colored dots whose x, y coordinates correspond to coordinates in the PCA basis. Once in the PCA basis a series of linear transformations are applied to the data. For improved clarity, we keep the data fixed throughout our algorithm visualization and represent linear transformations as change-of-basis (black arrows). Steps 2 and 3 of the algorithm find a new basis which more aptly captures spectral variability among stellar siblings, and Step 4 of the algorithm rescales basis vectors of this basis based on a comparison between their spectral flux variance among stellar siblings and among field stars. The outcome of the RSCA algorithm is a new representation of the spectra in which dimensions that are unhelpful for discriminating stellar siblings are minimized in amplitude (through a stretching-out of basis vectors). Conversely, dimensions that are helpful in recognizing stars within the same open clusters are made larger (through a squeezing of basis vectors). The K vector of principal components for each star can be collapsed into a measure of chemical similarity through a Euclidean distance measure between stars of their scaled representation, output from RSCA, where d = $\sqrt{{({n}_{K}-{n}_{K}^{{\prime} })}^{2}}$ for any pair of stars $n,n^{\prime}$.

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We now walk step-by-step through the successive linear transformations involved in the RSCA algorithm. For following along, the pseudo-code for RSCA is provided in Appendix D and the full source code of our project, which contains a Python implementation, is made available online. ⁴

In the first step of our approach, denoted as (1) "Compress spectra", in Figure 1, we apply PCA to X_pop to convert the population of stellar spectra into a lower-dimensional representation. This dimensionality-reduction step serves to make the algorithm more data efficient, which is crucial given the risk of overfitting from the small number of open clusters within our data set.

As some stellar bins are flagged as untrustworthy, we use PPCA, a variant of the PCA algorithm which can accommodate missing values. After finding principal components of X_pop, we compress the data by discarding all but the K-largest principal components, where K is a hyperparameter requiring tuning. Then, data sets X_pop and X_clust are each transformed using the K-basis vectors, which we call Z_pop and Z_clust, the representation of the spectra in the PCA basis of X_pop. These have shapes of ${N}_{{{\rm{d}}}_{\mathrm{pop}}}\times {N}_{{\rm{K}}}$ and ${N}_{{{\rm{d}}}_{\mathrm{clust}}}\times {N}_{{\rm{K}}}$.

Step (1) in our procedure of our PCA compression is a preprocessing step. The steps that follow fall into the realm of a general-purpose metric-learning algorithm. These rely on assumptions about the PCA-compressed spectra being satisfied. Performance should be robust to within small departures from these assumptions, but will still ultimately be tied to how well these assumptions are respected. We lay out our assumptions for steps (2)–(4) below.

3.3.1. Assumptions

3.3.2. Step 2: Sphering to Transform the Population Covariance Matrix into the Identity Matrix

3.3.3. Step 3: Reparameterization to Diagonalize the Cluster Covariance Matrix

3.3.4. Step 4: Scaling to Maximize Discriminative Power in Identifying Chemically Similar Stars

In the final step of the metric-learning algorithm, denoted by (4) "Rescale" in Figure 1, basis vectors are scaled proportionally to their dimension's usefulness at recognizing open clusters. This is done by applying separately to each dimension of the representation a scaling factor. Here, the independent scaling of dimensions is justified by Σ_clust and Σ_pop being diagonal covariance matrices.

We use the separate individual clusters within X_clust and field populations X_pop to measure variance in each dimension to determine our scaling factor. We design our scaling factor such that, after transformation, distances between pairs of random stars quantify the ratio between the probability that pairs of stars originate from the same (open) cluster and the probability that they do not. To put it another way, we seek to scale dimensions such that pairs of stars that are more likely to originate from the same cluster as compared to originating from different clusters have a smaller separation (i.e., Euclidean distance) in the representation than less-likely pairs.

Under our set of assumptions, along a dimension, i, among the K dimensions, random stellar siblings (intracluster stars) are distributed as ${z}_{\mathrm{clust}}\sim N({\mu }_{{\mathrm{clust}}_{{\rm{i}}}},{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}})$, where ${\mu }_{{\mathrm{clust}}_{{\rm{i}}}}$ and ${\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}$ are the mean and standard deviation along dimension i (at this stage in the algorithm). Accordingly, using the standard formula for the sum of normally distributed variables, the one-dimensional distance along i between pairs of random stellar siblings ${z}_{{{clust}}_{1i}}$ and ${z}_{{{clust}}_{2i}}$ follows a half-normal distribution, ${d}_{{\mathrm{clust}}_{{\rm{i}}}}\sim | {z}_{{{clust}}_{1i}}-{z}_{{{clust}}_{2i}}| =| N(0,2{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}})|$. Likewise, the distance, ${d}_{{\mathrm{pop}}_{{\rm{i}}}}$, between pairs of random field stars follows a similar half-normal distribution, ${d}_{{\mathrm{pop}}_{{\rm{i}}}}\sim | N(0,2{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}})|$, where ${\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}$ is the standard deviation among field stars along dimension i.

For a pair of stars observed a distance, d_i , away from each other along a dimension, i, the ratio between the probability of the pair originating from the same cluster (intracluster) and the probability of the pair not originating from the same cluster (intercluster) is

$$\begin{eqnarray}&&{r}_{i}({d}_{i})=\displaystyle \frac{p({d}_{{\mathrm{clust}}_{{\rm{i}}}}={d}_{i})}{p({d}_{{\mathrm{pop}}_{{\rm{i}}}}={d}_{i})}=\left|\displaystyle \frac{N(0,2{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}})}{N(0,2{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}})}\right|,\end{eqnarray} \tag{ 1 }$$

which evaluates to (as distances d_i are by design greater than 0)

$$\begin{eqnarray}&&{r}_{i}({d}_{i})={A}_{i}{e}^{\tfrac{-{d}_{i}^{2}}{2{\sigma }_{{r}_{i}}^{2}}},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{A}_{i}=\displaystyle \frac{{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}}{{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}}\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&{\sigma }_{{r}_{i}}=\displaystyle \frac{{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}}{\sqrt{{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}^{2}-{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}^{2}}}.\end{eqnarray} \tag{ 4 }$$

As dimensions are assumed to be independent, the probability ratio accounting for all dimensions is the product of the probability ratio of the separate dimensions:

$$\begin{eqnarray}&&r=\prod _{i=0}^{K}\displaystyle \frac{N(0,2{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}})}{N(0,2{\sigma }_{{\mathrm{pop}}_{{\rm{i}}}})}={{Ce}}^{-\tfrac{1}{2}{\sum }_{i=0}^{D}{\left(\tfrac{{d}_{i}}{{\sigma }_{{r}_{i}}}\right)}^{2}},\end{eqnarray} \tag{ 5 }$$

where $C={\prod }_{i=0}^{K}{A}_{i}$.

From this expression, it can be seen that multiplying dimensions by a scaling factor of $\tfrac{1}{{\sigma }_{{r}_{i}}}$ leads to a representation in which the probability ratio, r, is a function of Euclidean distance and where pairs of stars with smaller Euclidean distance separation have a higher probability of originating from the same open cluster as compared to their probability of originating from a different cluster than pairs with larger Euclidean separation.

It is clear that dividing dimensions by a scaling factor of $\tfrac{1}{{\sigma }_{{r}_{i}}}$ induces a representation where distances, d, which are measured between the scaled reconstructed representation, directly encode the probability of stars originating from the same cluster, as desired for our metric-learning approach. However, using this expression as a scaling factor requires evaluating the ${\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}$'s and ${\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}$'s along all dimensions. Because the representation has been sphered, the population's standard deviation is unity along all directions (${\sigma }_{{\mathrm{pop}}_{{\rm{i}}}}=1$). We estimate the intracluster standard deviations using a pooled variance estimator:

$$\begin{eqnarray}&&{\sigma }_{{\mathrm{clust}}_{{\rm{i}}}}^{2}=\displaystyle \frac{{\sum }_{j=1}^{k}\left({n}_{j}-1\right){\sigma }_{{ji}}^{2}}{{\sum }_{j=1}^{k}\left({n}_{j}-1\right)},\end{eqnarray} \tag{ 6 }$$

where ${\sigma }_{{ji}}^{2}$ refers to the sample variance along dimension, i, for the sample of stars belonging to an open cluster, j, containing n_j stars in X_clust. To make the algorithm more robust to the presence of any outliers in the data set, such as misclassified stellar siblings, we use the median absolute deviation (MAD) as an estimator for the sample standard deviation, σ_ji . That is,

$$\begin{eqnarray}&&\mathrm{MAD}=\mathrm{median}\left(\left|{X}_{i}-\tilde{X}\right|\right),\end{eqnarray} \tag{ 7 }$$

where X_i and $\tilde{X}$ are, respectively, the data values and median along a dimension.

To better understand the effect of our scaling factor on the representation it is applied to, it is instructive to look into the impact it has on the distances between stars along dimensions of a representation. When stars belonging to the same cluster have, along a dimension, a similar standard deviation to the full population of stars (i.e., σ_clust ≈ σ_pop), the dimension carries no information for recognizing cluster member stars and the scaling factor accordingly fully suppresses it σ_r → ∞ . On the other hand, for dimensions where the population's standard deviation, σ_pop, is significantly larger than the cluster standard deviation, σ_clust, the population's standard deviation is no longer relevant and σ_r ≈ σ_clust. That is to say, the scaling devolves into measuring distances relative to the number of standard deviations away from the cluster standard deviation.

For our experiments we use spectra from the public APOGEE DR16 (Ahumada et al. 2020) to create X_clust and X_pop, our data sets of field and open-cluster stars. Our field data set, X_pop, contains spectra for 151,145 red-giant-like stars matching a set of quality cuts on the sixteenth APOGEE data release described below. Our open-cluster data set, X_clust, contains spectra for 185 stars distributed across 22 open clusters, obtained after further quality cuts using the Open Cluster Chemical Abundance and Mapping (OCCAM) value-added catalog (Donor et al. 2020), a catalog containing information about candidate open clusters observed by APOGEE. We also create baseline data sets, Y and Y_clust, containing stellar abundances for the stars in X and X_clust. We include abundances for 21 species in Y and Y_clust: C, CI, N, O, Na, Mg, Al, Si, S, K, Ca, Ti, Ti II, V, Cr, Mn, Fe, Co, Ni, Cu, and Ce. These abundances are derived from the X_H entry in the allStar FITS file.

To create the data set of field stars. X_pop, we make the following data set cuts. With the intention of only preserving red-giant stars, we discard all but those stars for which 4000 < T_eff < 5000 K and $1.5\lt \mathrm{log}g\lt 3.0$ dex, where we use the T_eff and $\mathrm{log}g$ derived by the APOGEE Stellar Parameter and Chemical Abundances Pipeline (ASPCAP) pipeline. In addition, we further exclude any stars for which some stellar abundances of interest were not successfully estimated by the ASPCAP pipeline by removing any star containing abundances set to −9999.99 for any of our 21 species of interest. These abundance cuts are applied using the X_H and X_Fe ASPCAP fields rather than the named elemental abundance fields. This selection excludes an additional % of stars. We also exclude all spectra for which the STAR_BAD flag is set in ASPCAPFLAG. The pseudo-continuum spectra of those remaining stars, as found in the AspcapStar FITS file, were used to create the matrix, X_pop, in which each column contains the spectrum of one star. As described in Majewski et al. (2017), this pseudo-continuum normalization procedure involves a series of transformations that aim to standardize the stellar spectra. These involve subtracting the stellar continuum, shifting to the star's rest frame, removing interstellar reddening and atmospheric absorption effects, and normalizing the spectrum.

To create the data set of open-cluster member stars, X_clust, we cross-match our filtered data set with the OCCAM value-added catalog (Donor et al. 2020) so as to identify all candidate open clusters observed by APOGEE. We only keep those spectra of stars with open-cluster membership probability CG_PROB > 0.8 (Cantat-Gaudin et al. 2018). After this cross-match, we further filtered the data set by removing those clusters containing only a single member star as these are not useful for us. Additionally, we further discard one star with Apogee ID "2M19203303+3755558" found to have a highly anomalous metallicity. After this procedure, 185 OCCAM stars remain, distributed across 22 clusters. We do not cut any stars based on their signal-to-noise ratio. The stars in X_pop have a median signal-to-noise ratio of 157.2 and interquantile range of 102.0–272.4, while those in X_clust have a median signal-to-noise ratio of 191.4 and interquantile range of 117.7–322.6.

Because of cosmic rays, bad telluric line removal, or instrumental issues, the measurements for some bins of stellar spectra are untrustworthy. We censor such bad bins to prevent them from impacting our low-dimensional representation. Censored bins are treated as missing values in the PPCA compression. In this work, we have censored any spectral bin for which the error (as found in the AspcapStar FITS file error array) exceeds a threshold value of 0.05. Additionally, we censor for all stars in the data set those wavelength bins located near strong interstellar absorption features. More details about the model-free procedure for censoring interstellar features can be found in Appendix A.

Evaluating how good a representation is at measuring the chemical similarity of stars requires a goodness-of-fit indicator for assessing the validity of its predictions. We use the "doppelganger rate" as our indicator. This is defined as the fraction of random pairs of stars appearing as similar to, or more similar than, stellar siblings according to the representation, where similarity is measured in terms of distance, d, in the studied representation. It is worth noting that this procedure for estimating doppelganger rates is related to but different from the probabilistic approach presented in Ness et al. (2018).

We estimate doppelganger rates on a per-cluster basis by measuring distances between pairs of stars in the RSCA output representation. For each cluster in X_clust, the doppelganger rate is calculated as the fraction of pairs composed of one cluster member and one random star whose distance in the studied representation, d_{inter−family}, is less than the median distance amongst all cluster pairs, d_{intra−family}. That is, d_{intra−family} are pairs composed of two confirmed cluster members within X_clust and d_{inter−family} are pairs with one random field stars selected from X_pop and one cluster member from the studied cluster in X_clust. When calculating d_{inter−family}, we only consider pairs of stars with similar extinction and radial velocity, that is to say with ${\rm{\Delta }}\mathrm{AK}\_\mathrm{TARG}\lt 0.05$ and ΔVHELIO_AVG < 5. By only comparing stars at similar extinctions and similar velocities, we ensure that any model being investigated cannot reduce its doppelganger rate through exploiting extinction or radial velocity information in the spectra.

So as to facilitate comparisons between different representations, we aggregate the per-cluster doppelganger rates into a "global" doppelganger rate, which gives an overall measurement of a representation's effectiveness at identifying open clusters. The global doppelganger rate is obtained by averaging the per-cluster doppelganger rates through a weighted average in which clusters are weighted by their size in X_clust.

There is an added subtlety to assessing a representation through its global doppelganger rate. There are very few open-cluster stars in the data set. Therefore, RSCA as a data-driven procedure applied to open clusters is susceptible to overfitting to the open-cluster data set. To prevent overfitting from affecting results, we carry out a form of cross-validation in which clusters are excluded from the data set used for the derivation of their own doppelganger rate. In this scheme, calculating the global doppelganger rate of an RSCA representation requires repeated application of our algorithm, each time on a different subset with one cluster removed, as many times as there are open clusters.

We caution that our cross-validation approach has some implications on the derived doppelganger rates. Because every cluster's doppelganger rate is evaluated on a slightly different data subset, the quoted distances and doppelganger rates are not comparable from cluster to cluster.

The number of principal components used in the compression or encoding stage of RSCA (Step 1) is an important hyperparameter requiring tuning. In Figure 2, we plot the doppelganger rate against the number of principal components, both with and without using the cross-validation procedure described in Section 4.2. Results without cross-validation display significant overfitting and are mostly shown in an effort to highlight the importance of the cross-validation procedure.

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This figure illustrates how, unsurprisingly, RSCA's performance is strongly dependent on the PCA dimensionality. Doppelganger rates decrease with increasing PCA dimensionality, up until a dimensionality of size 30. At K > 30, doppelganger rates start increasing because of overfitting. Since on our studied data set, RSCA reaches its peak performance for 30 PCA components (compared to 7514 bins in the raw spectral representation), all further quoted results and figures will accordingly use the first 30 PCA components.

That RSCA's performance still improves up to a dimensionality of 30 is interesting. This demonstrates that a hyperplane of at least size 30 is required for capturing the intrinsic variations of APOGEE stellar spectra, a number noticeably larger than the 10-dimensional hyperplane found in Price-Jones & Bovy (2017). Methodological differences between studies may partially explain such differences. For example, the PCA fit in Price-Jones & Bovy (2017) was applied on spectra displaying limited instrumental systematics and with nonchemical imprints on the spectra preliminary removed. It should also be noted that this does not mean that the chemical space is 30-dimensional. Some PCA dimensions may capture instrumental systematics or nonchemical factors of variation, such as residual sky absorption and emission and interstellar dust imprints. Also, since chemical species leave nonlinear imprints, because of the linearity of PCA, each nonlinear chemical dimension may require multiple PCA components to be fully captured.

We now study which spectral features are leveraged by the RSCA algorithm when recognizing open clusters. In the RSCA rescaled basis (Step 4), the dimensions are scaled proportionally to their perceived usefulness at measuring chemical similarity. Therefore, factors of variation judged most important by RSCA will correspond to the most strongly scaled dimensions of the representation (i.e., with the largest $\tfrac{1}{{\sigma }_{{r}_{i}}}$). Figure 3 shows the relationship between the three features with scaled dimensions with the largest amplitudes and [Fe/H] for a representation obtained by running the RSCA algorithm with a PCA dimensionality of 30.

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As seen from the left-most panel, there is a close relationship between "Feature #1", the RSCA dimension with the largest associated scaling factor, and the ASPCAP [Fe/H] label. The relationship is close to a one-to-one mapping, which illustrates how this feature traces the metallicity content of the stellar spectra. Because "Feature #1" (as a direction in a hyperplane of stellar spectra) is a linear function of the stellar spectra and metallicity is a nonlinear feature, some degree of scatter in the relationship is expected.

The relationship between "Feature #2" and [Fe/H] (second panel) exhibits the same bimodality as observed when plotting α enhancements [α/Fe] against [Fe/H] (Leung & Bovy 2018). This indicates that "Feature #2" captures α-element enhancements. It is particularly noteworthy that we are able to recover the α-element bimodality when the open clusters in our data set (orange markers) are located only in the low-α sequence. This demonstrates the metric model's capacity to extrapolate to abundance values outside the range of values covered in the open-cluster training data set. This provides evidence that the model may still be effective for stars atypical of those in the open-cluster data set.

The relationship between "Feature #3" and metallicity (third panel) is not as easily interpreted. Given the nonlinear nature of metallicity, it is possible that it encodes residual metallicity variability not captured by "Feature #1", but it is also possible that it contains some further independent chemical dimension.

This figure illustrates a nice property of RSCA. Because dimensions of the RSCA representation correspond to eigenvectors of the covariance matrix, Σ_clust, the RSCA algorithm, at least to first order, assigns the distinct factors of variation within spectra to separate dimensions of the representation. That is to say, that the dimensions of the RSCA representation capture distinct factors of variation, such as the metallicity or the α-element abundance, rather than a combination of factors of variation. Additionally, the most important factors of variation for recognizing open clusters occupy the dimensions with the largest scaling factors. This property makes RSCA particularly versatile. For example RSCA can be used to separate out high- and low-α-abundance stars in the disk, or to select low-metallicity stars. Additionally, it is likely that because of this property RSCA could be used to search for hidden chemical factors of variation within stellar spectra, although this has not been attempted in this paper.

We found that some of the dimensions of the RSCA representation showed trends with radial velocity. An example of a dimension showing a trend with radial velocity is shown in the last panel of Figure 3 and an investigation into the detailed causes of the radial velocity trends is presented in Appendix B. The existence of such trends indicates that, even after the ASPCAP pseudo-continuum normalization procedure which shifts spectra to the rest frame, RSCA is still capable, at least weakly, to exploit radial velocity information in the spectra to recognize stellar siblings. Because only a subset of the dimensions shows such trends, a representation tracing only chemistry can be obtained by only keeping those dimensions which show no trends with radial velocity. In this work, we propose to only keep the first three dimensions of the representation. While this choice might appear particularly stringent, as we will show in coming sections, these three dimensions contain the bulk of the discriminative power of the representation (see Table 1).

In this section, we compare the effectiveness of measuring chemical similarity using a data-driven approach on the spectra, with that achievable from using measured stellar abundances. To do so, we compare the doppelganger rates that are obtained by RSCA to those from using stellar-abundance labels. The results of such a comparison are shown in Figure 4. For this figure, doppelganger rates are measured consistently from abundances and the RSCA approach (see figure caption for more detail), such that any differences in performance can be attributed to underlying differences in the information content of the representations. For this comparison, we omit the PCA dimensionality-reduction step when working with abundances, but otherwise apply the exact same algorithmic approach to abundances and to the spectra. We also compare the performance of the RSCA metric-learning approach (shown in red) to that of alternative, simpler representation rescaling approaches applicable for recognizing open clusters from abundances (shown in blue and green). We remind the reader that the doppelganger rates are evaluated for pairs of stars at the same extinctions and radial velocities. This guarantees that the doppelganger rate cannot be artificially reduced through our model exploiting information relating to radial velocity or extinction. Per-cluster doppelganger rates are also provided in Appendix E.

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From this figure, we see that executing all steps of the RSCA is crucial for obtaining low doppelganger rates when working directly with spectra. This is seen from how the low doppelganger rates are only obtained with full application of our metric-learning approach (red). On the other hand, when working with stellar abundances, the RSCA approach appears to bring only limited benefits, as seen from the only slight difference in doppelganger rates between measuring distances in the raw abundance space (blue) and in the transformed space (red) that is obtained by application of our full metric-learning approach minus the PCA compression. This result is not surprising and reflects how most steps of the RSCA approach are designed with the aim of generating a representation comparable to stellar labels, that is to say, one where all factors of variation other than chemical factors of variation are removed. The RSCA approach does, however, still bring some benefits when working with abundances as seen from the lower doppelganger rates.

This figure also shows that excluding chemical species improves the doppelganger rate. This is seen from how the doppelganger rate is lower when using a carefully chosen subset of species (right) than when using the full set of abundances (center). This can appear counterintuitive as it implies that more data leads to worsened performance, but in this specific case, where the uncertainties on abundances are not accounted for, it can be justified by the low intrinsic dimensionality of chemical space. Since many species contain essentially the same information, adding species with higher uncertainty into the representation adds noise into the representation. It does this without contributing any additional information beneficial for recognizing open clusters. We expect that such an effect would disappear when accounting for uncertainties on stellar labels, but it is still a good illustration of the brittleness of abundance-based chemical tagging.

The combination of species shown in red is the set of species which were found, after manual investigation, to yield the lowest doppelganger rates. This is the combination of stellar individual element abundance labels Fe, Mg, Ni, Si, Al, C, and N (with respect to Fe with the exception of Fe, which is with respect to H). The doppelganger rate from this combination of species, of 0.023, despite being the smallest doppelganger rate achieved from stellar labels, is higher than the doppelganger rate obtained from stellar spectra of 0.020 (2%). That our method is able to produce better doppelganger rates from spectra than from stellar labels highlights the existence of information within stellar spectra not being adequately captured by stellar labels. While stellar labels are derived from synthetic spectra which only approximately replicate observations, our fully data-driven model makes direct use of the spectra translating into lower doppelganger rates.

That the PCA representation is 30-dimensional does not mean that all 30 dimensions carry information useful for recognizing open clusters. To get a grasp of the dimensionality of the chemical space captured by RSCA, we calculated the doppelganger rates for RSCA representations in which only the dimensions with the largest scaling factors are kept (ie other dimensions are excluded from the doppelganger rate distance calculations). We calculate doppelganger rates multiple times, each with a different number of dimensions preserved. The results of this investigation are shown in Table 1. From this table, we see that the dimensionality of spectra appears to be, at least to first degree, extremely low. The top two dimensions of the RSCA model (as shown in Figure 3) are capable of matching the performance obtained from stellar labels while the top four dimensions exceed the performance from using the full representation. The four-dimensional representation is even more effective at recognizing chemically identical stars than the full RSCA representation, which itself was more effective than stellar labels. It is not a new result that the dimensionality of chemical space probed by APOGEE is low. Recent research suggest that, at the precisions captured by APOGEE labels, chemical abundances live in a very low-dimensional space for stars of the disk. For example, it was found in Ness et al. (2019) and Ting & Weinberg (2021) that [Fe/H] and stellar age or [Fe/H] and [Mg/Fe] could predict all other elemental abundances to within, or close to, measurement precision (nonetheless, Ting & Weinberg 2021 and Weinberg et al. 2021 argue that correlations of abundance residuals imply underlying intrinsic structure in abundance space, even if the scatter of these residuals is only moderately larger than the per-star observational uncertainties). However, while previous analyses have depended on abundances to show this, here we can do this directly from spectra. As our methodology directly picks up on factors of variation and, if not controlled for, is capable of picking up on weak factors of variation such as diffuse interstellar bands, we can be confident that any remaining chemical factors of variation are either (i) highly nonlinear for our model to not be capable of picking up on them, (ii) very weak spectral features, or (iii) not particularly discriminative of open clusters, as would be the case for chemical variations arising from internal stellar processes, for example induced by accretion of planetary materials or internal stellar processes.

Our method learns to measure chemical similarity directly from open clusters without reliance on external information. Because of this, its performance will be tightly linked to the quality and quantity of data available. Figure 5 attempts to estimate our method's dependency on the size of the open-cluster data set. In this figure, for varying PCA dimensionalities, we plot the expected doppelganger rate for an open-cluster data set containing a given quantity of open clusters, whose number is given by the x-axis. We estimate the expected doppelganger rate for a given number of open clusters by estimating and averaging the doppelganger rates for all data subsets containing that number of clusters. From this figure, we see that the larger PCA dimensionalities still benefit from the addition of open clusters. This is suggestive that performance would likely further improve with access to additional open clusters. Additionally, we may also expect the addition of stars to open clusters to also improve the doppelganger rates. Such larger data sets may also enable the usage of more complex nonlinear metric-learning approaches, which may yield further improvements not captured in this figure.

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